Study on Local-Structure Symmetrization of K2XF6 Crystals Doped with Mn4+ Ions by First-Principles Calculations

The crystals of Mn4+-activated fluorides, such as those of the hexafluorometallate family, are widely known for their luminescence properties. The most commonly reported red phosphors are A2XF6: Mn4+ and BXF6: Mn4+ fluorides, where A represents alkali metal ions such as Li, Na, K, Rb, Cs; X=Ti, Si, Ge, Zr, Sn, B = Ba and Zn; and X = Si, Ge, Zr, Sn, and Ti. Their performance is heavily influenced by the local structure around dopant ions. Many well-known research organizations have focused their attention on this area in recent years. However, there has been no report on the effect of local structural symmetrization on the luminescence properties of red phosphors. The purpose of this research was to investigate the effect of local structural symmetrization on the polytypes of K2XF6 crystals, namely Oh-K2MnF6, C3v-K2MnF6, Oh-K2SiF6, C3v-K2SiF6, D3d-K2GeF6, and C3v-K2GeF6. These crystal formations yielded seven-atom model clusters. Discrete Variational Xα (DV-Xα) and Discrete Variational Multi Electron (DVME) were the first principles methods used to compute the Molecular orbital energies, multiplet energy levels, and Coulomb integrals of these compounds. The multiplet energies of Mn4+ doped K2XF6 crystals were qualitatively reproduced by taking lattice relaxation, Configuration Dependent Correction (CDC), and Correlation Correction (CC) into account. The 4A2g→4T2g (4F) and 4A2g→4T1g (4F) energies increased when the Mn-F bond length decreased, but the 2Eg → 4A2g energy decreased. Because of the low symmetry, the magnitude of the Coulomb integral became smaller. As a result, the decreasing trend in the R-line energy could be attributed to a decreased electron–electron repulsion.


Introduction
Incandescent and fluorescent lighting sources have been rapidly replaced by White Light Emitting Diodes (WLEDs) in homes, offices, and public areas. They are made of a blue LED chip with a yellow phosphor. WLED is the most energy-efficient conversion source compared to previously existing lighting sources, yet it creates pseudo-white light due to a lack of red emissions. Rather than employing all basic colors of LED chips, mixing blue LED chips with yellow and red phosphors is simpler and less expensive. The blue LED chips are typically made of InGaN [1], whereas the yellow phosphor components are composed of Y 3 Al 5 O 12 : Ce 3+ [2]. The high-performance red phosphors are Eu 2+ doped nitrides [3][4][5][6][7][8][9]. Unfortunately, red phosphors are expensive due to scarcity and challenging synthesis conditions, such as extreme temperatures and nitrogen pressure. Finding novel red phosphor materials that are appropriate for WLED is currently challenging. Significant performance factors for white light that are used in general lighting include high Quantum Efficiency (QE > 70%), resistance to thermal quenching (preferably > 80% of the luminescence intensity should be sustained at 450 K), and strong color quality, which includes a low Correlated Color Temperature (CCT) of 3000 K and a high Color Rendering Index (CRI > 70).
The most commonly reported red phosphors are fluoride-based, such as A 2 XF 6 : Mn 4+ and BXF 6 : Mn 4+ , where A represents alkali metal ions, such as Li, Na, K, Rb, Cs; X = Ti, Si, Ge, Zr, Sn), B = Ba and Zn and X = Si, Ge, Zr, Sn, and Ti. K 2 SiF 6 : Mn 4+ , KNa 2 SiF 6 : Mn 4+ , and K 2 TiF 6 : Mn 4+ , in particular, have shown good potential for WLED as a red phosphor under blue LED chip stimulation. The first red Mn 4+ -doped fluoride phosphor, K 2 SiF 6 : Mn 4+ , was published in 1973 [10]. K 2 SiF 6 is one of the most promising hexafluoride hosts, with a slightly higher Luminous Efficacy of Radiation (LER) upon Mn 4+ doping than K 2 TiF 6 and a 30% higher External Quantum Efficiency (EQE) than KnaSiF 6 : Mn 4+ [11]. Mn 4+ , when doped in K 2 SiF 6 or K 2 TiF 6 as a red phosphor, yields WLEDs with warm-white CCTs 3000 K and good CRIs~90, as demonstrated by Setlur et al. [12]. The d-d transitions in Mn 4+ cause the particular red emission line detected in K 2 SiF 6 : Mn 4+ to be approximately 630 nm (15,873 cm −1 or 1.97 eV) [13]. Nevertheless, the chemical and thermal stability problems and safety hazards of K 2 SiF 6 and K 2 TiF 6 doped with Mn 4+ have been reported. The aforementioned red-phosphor performance is highly dependent on local structure. Numerous research teams have concentrated on the modification and enhancement of phosphor luminescence properties through the alteration of the local crystal structure. The "Cation-Size-Mismatch effect", "Neighboring-Cation Substitution effect", and "Nanosegregation and Neighbor-Cation Control effect", among other new luminescence mechanisms, were reported by Liu's group in Ce 3+ and Eu 2+ -doped (oxy)nitrides based on the variation in the local crystal structure [14][15][16][17]. Ram's team also showed that slight modifications to the local structure of phosphor systems such as La 3 xCexSi 6 N 11 , SrxBa 2 xSiO 4 : Eu 2+ , etc., could lead to appreciable gains in luminescence performance [18,19]. Cheetham's team discovered that local crystal structural deformation accounted for a significant spectrum change from blue to yellow light from Ca 2 SiO 4 : Ce [20].
The Ligand Field Theory (LFT) has been frequently used to successfully evaluate the multiplet energy levels and optical spectra of Transition Metal (TM) ions in crystals [21]. However, it is an empirical method in which the measured spectrum is used to determine the Racah parameters and crystal field splitting. Watanabe and Kamimura produced the first non-empirical forecast in the late 1980s [22,23] using a combination of the local density approximation (LDA) and LFT. On the other hand, a number of teams, including Daul et al. [24], Wissing et al. [25], and Oliveira et al. [26,27], have also performed first-principle calculations based on the Density Functional Theory (DFT). However, obtaining the manyelectron wave functions proved unfeasible. During the previous ten years, Ogasawara's team created the Discrete Variational Multi-Electron (DVME) approach [28]: a non-empirical first-principles many-electron calculation technique. It uses both a Configuration Interaction (CI) computation and DFT. DVME consists of two phases. To begin, one-electron Molecular Orbital (MO) calculations are performed using the Discrete Variational Xα (DV-Xα) method. The CI method is then used to perform many-electron computations, which is the main stage of the DVME approach. It has been shown that DVME is a powerful tool for estimating absorption spectra, energy levels, transition energies, etc., without the use of any empirical parameters.
Up until recently, there has been no study on the influence of local structure on symmetry (switching from a high-symmetry to a low-symmetry configuration) or on the luminous qualities of red phosphors. Therefore, the goal of this research was to investigate the effect of local structural symmetrization on the polytypes of K 2 XF 6 crystals, namely O h -K 2 MnF 6 , C 3v -K 2 MnF 6 , O h -K 2 SiF 6 , C 3v -K 2 SiF 6 , D 3d -K 2 GeF 6 , and C 3v -K 2 GeF 6 . The DVME method was used to calculate their multiplet energy levels.

Materials and Methods
Polytypes of various K 2 XF 6 crystals were used to create seven-atom model clusters. The cubic K 2 MnF 6 ICSD #47213 had a = 8.221 lattice parameter, a space group Fm3m, and O h symmetry [29]. The lattice parameters of hexagonal K 2 MnF 6 ICSD #60417 were a = 5.719 Å and c = 9.330 Å, with space group P63mc and C 3v Symmetry [30]. The cubic K 2 SiF 6 ICSD #2940 had a= 8.134 lattice parameters, a space group Fm3m, and O h symmetry [31]. The lattice parameters of hexagonal K 2 SiF 6 ICSD #158483 were a = 5.6461 Å and c = 9.2322 Å, with the space group P63mc and C 3v Symmetry [32]. The lattice parameters of rhombohedral K 2 GeF 6 ICSD #24026 were a = 5.63 and c = 4.66, with the space group P3m1 and D 3d Symmetry [33]. The lattice parameters of hexagonal K 2 GeF 6 ICSD #30310 were a = 5.71 Å and c = 9.27 Å, with the space group P63mc and C 3v Symmetry [34]. The computations were performed using O h , D 3d , and C 3v symmetry for clusters built from K 2 XF 6 (X = Mn, Si, or Ge) and crystals with cubic, rhombohedral, and hexagonal structures, respectively. Figure 1a-c depicts the various types of crystal structures of the materials under consideration, including namely cubic, rhombohedral, and hexagonal structures. Figure 1d-f were model clusters made up of seven atoms, one X 4+ ion surrounded by 6 F − . Here, we adopted the results of the Mn K-edge Extended X-ray Absorption Fine Structure (EXAFS) measurement of K 2 XF 6 (X = Si, or Ge): Mn 4+ [35]. The Mn-F bond lengths for K 2 SiF 6 : Mn 4+ and K 2 GeF 6 : Mn 4+ were 1.807 and 1.810 Å, respectively. The one-electron calculations utilizing the DV-Xα method were then carried out [36][37][38]. The DVME approach was used to account for the many-electron effects [28]. The energy corrections such as Configuration Dependent Correction (CDC) and Correlation Correction (CC) were also considered. Racah parameters were used to calculate the Coulomb integrals as well. These methods' specific steps are described in Reference [35]. Variational Xα (DV−Xα) method. The CI method is then used to perform many−electron computations, which is the main stage of the DVME approach. It has been shown that DVME is a powerful tool for estimating absorption spectra, energy levels, transition energies, etc., without the use of any empirical parameters. Up until recently, there has been no study on the influence of local structure on symmetry (switching from a high−symmetry to a low−symmetry configuration) or on the luminous qualities of red phosphors. Therefore, the goal of this research was to investigate the effect of local structural symmetrization on the polytypes of K2XF6 crystals, namely The DVME method was used to calculate their multiplet energy levels.

Materials and Methods
Polytypes of various K2XF6 crystals were used to create seven−atom model clusters. The cubic K2MnF6 ICSD #47213 had a = 8.221 lattice parameter, a space group 3 , and Oh symmetry [29]. The lattice parameters of hexagonal K2MnF6 ICSD #60417 were a = 5.719 Å and c = 9.330 Å, with space group 63 and C3v Symmetry [30]. The cubic K2SiF6 ICSD #2940 had a= 8.134 lattice parameters, a space group 3 , and Oh symmetry [31]. The lattice parameters of hexagonal K2SiF6 ICSD #158483 were a = 5.6461 Å and c = 9.2322 Å, with the space group 63 and C3v Symmetry [32]. The lattice parameters of rhombohedral K2GeF6 ICSD #24026 were a = 5.63 and c = 4.66, with the space group 3 1 and D3d Symmetry [33]. The lattice parameters of hexagonal K2GeF6 ICSD #30310 were a = 5.71 Å and c = 9.27 Å, with the space group 63 and C3v Symmetry [34]. The computations were performed using Oh, D3d, and C3v symmetry for clusters built from K2XF6 (X = Mn, Si, or Ge) and crystals with cubic, rhombohedral, and hexagonal structures, respectively. Figure 1a-c depicts the various types of crystal structures of the materials under consideration, including namely cubic, rhombohedral, and hexagonal structures. Figure 1d-f were model clusters made up of seven atoms, one X 4+ ion surrounded by 6 F − . Here, we adopted the results of the Mn K−edge Extended X−ray Absorption Fine Structure (EXAFS) measurement of K2XF6 (X = Si, or Ge): Mn 4+ [35]. The Mn−F bond lengths for K2SiF6: Mn 4+ and K2GeF6: Mn 4+ were 1.807 and 1.810 Å, respectively. The one−electron calculations utilizing the DV−Xα method were then carried out [36][37][38]. The DVME approach was used to account for the many−electron effects [28]. The energy corrections such as Configuration Dependent Correction (CDC) and Correlation Correction (CC) were also considered. Racah parameters were used to calculate the Coulomb integrals as well. These methods' specific steps are described in Reference [35].

Bond Lengths
The Mn-F bond lengths of O h -K 2 MnF 6 , C 3v -K 2 MnF 6 , O h -K 2 SiF 6 , C 3v -K 2 SiF 6 , D 3d -K 2 GeF 6 , and C 3v -K 2 GeF 6 . are shown in Table 1. All six bond lengths are represented by letters d1, d2, d3, d4, d5, and d6, respectively. When the lattice relaxation effect was not used, the lengths of the Mn-F bonds dropped from O h -K 2 MnF 6 to C 3v -K 2 MnF 6 . This was similar to the trend for O h -K 2 SiF 6 : Mn 4+ to C 3v -K 2 SiF 6 : Mn 4+ . On the other hand, the trend for D 3d -K 2 GeF 6 : Mn 4+ to C 3v -K 2 GeF 6 : Mn 4+ was reversed. When the lattice relaxation effect was used, however, the Mn-F bond lengths decreased in all situations.  Figure 2 depicts the molecular orbital energies of O h -K 2 MnF 6 , C 3v -K 2 MnF 6 , O h -K 2 SiF 6 , C 3v -K 2 SiF 6 , D 3d -K 2 GeF 6 , and C 3v -K 2 GeF 6 . The Valence Band (VB) is represented by black solid lines. The Conduction Band (CB) is shown by the black dashed lines. The impurity levels are represented as t 2g and e g , with solid red and dashed blue lines, respectively. The tops of the VBs were set to zero. For O h -K 2 MnF 6 and C 3v -K 2 MnF 6 , the crystal field splitting (10Dq, defined as the differential energy between t 2g and e g levels) was estimated to be 1.79 and 2.68 eV, respectively. Without accounting for the lattice relaxation effect, the 10Dq of O h -K 2 SiF 6 and C 3v -K 2 SiF 6 were estimated to be 3.52 and 3.44 eV, respectively. They fell to 2.63 and 2.53 eV when the lattice relation effect was taken into account. In the case of D 3d -K 2 GeF 6 and C 3v -K 2 GeF 6 , the 10Dq was determined to be 2.76 and 2.72 eV, respectively. After accounting for the lattice relaxation effect, they fell to 2.52 and 2.61 eV, respectively.

Multiplet Energy Levels
Since the d-d transitions of K2XF6: Mn 4+ was prohibited by the parity selection rul the transition probabilities could not be determined. As a result, this report is restricted t energy levels. We estimated the doublet states 2 E,. 2 T2, and 2 T1, as well as the quartet state 4 T2 and 4 T1a. The absorption transitions start from the ground 4 A2 state to 4 T2 and 4 T1a state which often appeared as wide bands and were referred to as the U− and Y−band, respec tively. On the other hand, the emission transition started from the 2 E state to the groun 4 A2 state, which generally appeared as a sharp line and was referred to as R−line.
The pure K2MnF6 and K2SiF6: Mn 4+ computed multiplet energy diagrams with Oh an C3v symmetry are shown in Figure 3. A few adjustments, including CDC, CC, and lattic relaxation, were also assessed. Figure 3 demonstrates that quite often, the doublet state decreased when reduced symmetry was employed. Furthermore, CDC−CC correction ha a smaller impact on Oh−K2MnF6 than it did on C3v−K2MnF6, suggesting that C3v−K2MnF benefited more from correlation correction. On the other hand, the quartet states increase for pure K2MnF6 from Oh to C3v while they dropped for K2SiF6: Mn 4+ in the same orde This was expected because the Mn−F bond length, which varied widely depending on th material, primarily affected the quartet states.

Multiplet Energy Levels
Since the d-d transitions of K 2 XF 6 : Mn 4+ was prohibited by the parity selection rule, the transition probabilities could not be determined. As a result, this report is restricted to energy levels. We estimated the doublet states 2 E, 2 T 2, and 2 T 1, as well as the quartet states 4 T 2 and 4 T 1a . The absorption transitions start from the ground 4 A 2 state to 4 T 2 and 4 T 1a states which often appeared as wide bands and were referred to as the U-and Y-band, respectively. On the other hand, the emission transition started from the 2 E state to the ground 4 A 2 state, which generally appeared as a sharp line and was referred to as R-line.
The pure K 2 MnF 6 and K 2 SiF 6 : Mn 4+ computed multiplet energy diagrams with O h and C 3v symmetry are shown in Figure 3. A few adjustments, including CDC, CC, and lattice relaxation, were also assessed. Figure 3 demonstrates that quite often, the doublet states decreased when reduced symmetry was employed. Furthermore, CDC-CC correction had a smaller impact on O h -K 2 MnF 6 than it did on C 3v -K 2 MnF 6 , suggesting that C 3v -K 2 MnF 6 benefited more from correlation correction. On the other hand, the quartet states increased for pure K 2 MnF 6 from O h to C 3v while they dropped for K 2 SiF 6 : Mn 4+ in the same order. This was expected because the Mn-F bond length, which varied widely depending on the material, primarily affected the quartet states.
The predicted multiplet energy diagrams of K 2 GeF 6 : Mn 4+ with D 3d and C 3v symmetry are shown in Figure 4. CDC, CC, and lattice relaxation were also evaluated, similar to Figure 3. These findings showed that the average doublet state values for the two clusters were remarkably similar. Low symmetry was also found to have an impact on multiplet splitting. While the splitting of the 4 T 2 state decreased, it increased for the 4 T 1a , 2 T 2, and 2 T 1 states.  The predicted multiplet energy diagrams of K2GeF6: Mn 4+ with D3d and C3v symme are shown in Figure 4. CDC, CC, and lattice relaxation were also evaluated, similar to Fig  3. These findings showed that the average doublet state values for the two clusters w remarkably similar. Low symmetry was also found to have an impact on multiplet splitt While the splitting of the 4 T2 state decreased, it increased for the 4 T1a, 2 T2, and 2 T1 states. Figure 3. Pure K 2 MnF 6 and K 2 SiF 6 : Mn 4+ multiplet energy diagrams. Additionally, demonstrated is the impact of CDC, CC, and lattice relaxation. The left side of each column explains the calculation using O h -symmetric clusters, while the right side describes the calculation using C 3v -symmetric clusters. Black and red lines denote the doublet and quartet states, respectively. When the lower symmetry (C 3v ) was used, these states were further divided into the a (dashed lines) and e (solid lines) categories. There are the doublet states 2 E,. 2 T 2, and 2 T 1, as well as the quartet states 4 T 2 and 4 T 1a . The 4 A 2 is the ground state. The absorption occurred during the electronic transitions from the ground 4 A 2 state to 4 T 2 and 4 T 1a states (U-and Y-band, respectively), as illustrated by the green and blue arrows. The emission, on the other hand, happened as an electronic transition from the 2 E state to the ground 4 A 2 state (R-line), as illustrated by the red arrow. More information can be found in the text.

Coulomb Integrals
The Coulomb integrals of pure K 2 MnF 6 , K 2 SiF 6 : Mn 4+ , and K 2 GeF 6 : Mn 4+ are shown in Table 2. When low symmetry was used, the effective Coulomb integrals J eff estimated by cλJ AO almost always decreased. Although the J eff(t2g) of K 2 GeF 6 : Mn 4+ without lattice relaxation was greater than that of D 3d -K 2 GeF 6 : Mn 4+ , its tendency improved when lattice relaxation was considered. These findings suggest that reduced symmetry resulted in a smaller Coulomb integral. As a result, the decreasing trend of R-line energy could be attributed to a decreased electron-electron repulsion.
the other hand, happened as an electronic transition from the E state to the ground A2 state (R−line), as illustrated by the red arrow. More information can be found in the text.
The predicted multiplet energy diagrams of K2GeF6: Mn 4+ with D3d and C3v symmetry are shown in Figure 4. CDC, CC, and lattice relaxation were also evaluated, similar to Figure  3. These findings showed that the average doublet state values for the two clusters were remarkably similar. Low symmetry was also found to have an impact on multiplet splitting. While the splitting of the 4 T2 state decreased, it increased for the 4 T1a, 2 T2, and 2 T1 states.  Figure 4. K 2 GeF 6 : Mn 4+ multiplet energy diagrams. Additionally demonstrated is the impact of corrections, including CDC, CC, and lattice relaxation. A calculation using clusters with D 3d symmetry is described on the left side of each column, while a calculation using clusters with C 3v symmetry is described on the right side. The O h symmetry notations, in this instance, were borrowed. Black and red lines denote the doublet and quintet states, respectively; dashed (a level) and solid lines (e level) denote the multiplet splitting. There are the doublet states 2 E,. 2 T 2, and 2 T 1, as well as the quartet states 4 T 2 and 4 T 1a . The 4 A 2 is the ground state. The absorption occurred during the electronic transitions from the ground 4 A 2 state to 4 T 2 and 4 T 1a states (U-and Y-band, respectively), as illustrated by the green and blue arrows. The emission, on the other hand, happened as an electronic transition from the 2 E state to the ground 4 A 2 state (R-line), as illustrated by the red arrow. More information can be found in the text.  6 2− model clusters with O h , D 3d , and C 3v symmetry, Coulomb integral (eV) for the pure TM-3d atomic orbitals (J AO ) and the molecular orbitals (J MO ) were calculated. The adjustments were contrasted, including those with and without lattice relaxation. The orbital deformation parameter (λ) and the correlation correction factor (c) were multiplied by J AO to calculate the effective Coulomb integrals (J eff ).

Discussion
LEDs are now used in a variety of commonplace applications, including display backlights for smartphones, tablets, and televisions, as well as warm-white LEDs for energy-efficient lighting. A portion of the blue light from the LED chip is converted into white light by color-converting luminescent materials. This is accomplished by using doped wide-bandgap materials, also referred to as phosphors or Colloidal Quantum Dots (QDs). The color quality of white LEDs is improved when red-emitting phosphors are added when compared to the prototype's arrangement of a blue LED and a yellow Y 3 Al 5 O 12 : Ce 3+ . These luminous materials have an incredibly high luminescence efficiency, especially at room temperature and above, due to the involvement and stimulation of thermal phonons.
The requirements for a phosphor to be suitable for LED applications were described by Smet et al. [39] in detail. According to the Color Quality Scale (CQS) [40] or the CRI [41], the resulting white light source had a high color rendering. This was significant for illumination. For display applications to produce a broad color spectrum or high color purity, saturated colors were necessary. The lower the filtering losses, the better the phosphors' emission spectrum fits the color filters. Second, a phosphor must have a high LER, which is a metric for the average eye sensitivity of the spectrum, measured in lm/W) and a high Internal Quantum Efficiency (IQE), which is stable at high temperatures. Third, there needs to be significant blue light absorption, which raises the EQE. A phosphor can only be considered a serious candidate for LED applications when all four requirements are satisfied simultaneously.
The Mn 4+ emission center prefers to remain in the octahedral or modified octahedral position of the host due to the large ligand-field stabilizing energy of Mn 4+ in the six-fold coordination. In the initial LFT simulation, only the octahedral (O h ) crystal field was taken into account [42]. The doubly degenerate e g level had +6Dq more energy than the fivefold degenerate 3d level, and the triply degenerate t 2g level had −4Dq more energy. The intensity of the crystal field, Dq, changed based on the ion-crystal combination, and as a result, the crystal field splitting was 10Dq. Through the use of the electron-electron repulsion parameters A, B, and C, also referred to as Racah parameters, the impact of covalency could also be taken into account in this situation. The crystal field strength Dq, the Racah parameters B and C, and the multiplet energy levels E i could thus be used to explain them. The recognized Tanabe Sugano diagrams [21,43], which depict E i /B as functions of Dq/B for a fixed value of C/B, describe the energy levels of all d N systems in an octahedral crystal field as functions of Dq. The spectral characteristics of red phosphor materials were also described using the absorption and emission spectra. There were certain doublet states, such as 2 E, 2 T 1 , etc., and some quartet states, such as 4 A 2 , 4 T 2 , 4 T 1a , 4 T 1b , etc., since Mn 4+ -doped compounds contained three electrons filling ten degenerate 3d orbitals (3d 3 ). The energy was lowest in the ground state ( 4 A 2 ). The transitions from 4 A 2 to 4 T 2 (U-band) and 4 T 1a (Y-band) were utilized for absorption, while the transition from 2 E to 4 A 2 (R-line) was employed for emission when used as red phosphor materials.
Previously, we studied the potential of oxide and fluoride materials for red phosphor materials in WLED by DV-Xα and DVME methods. The investigation included lattice relaxation, orbital energy, multiplet energy, absorption spectra, energy correction, pressure dependence, and emitted light utilizing CIE 1931 color space [35,[44][45][46][47][48][49][50][51][52]. Because of the length of the Mn-F bond, the quartet state energies typically had a strong relationship with crystal field splitting. On the other hand, doublet state energies strongly depended on the correlation correction. The computational conditions to reproduce those optical properties depend on the material itself.
The study of low-structure symmetrization was required when understanding the properties of novel phosphor materials. According to our findings, a low structure had a substantial effect on multiplet structures, which could affect their performance as phosphor materials. Table 1 indicates that considering the lattice relaxation effect caused by the Mn 4+ , substitution resulted in a considerable shift in the bond lengths. The crystal field splittings were approximated using the one-electron DV-Xα approach, as shown in Figure 2. The 10Dq crystal field splitting tendency was caused by the lengths of the Mn-F bonds. Furthermore, the multiplet energies in Figures 3 and 4 were determined using the many-electron DVME approach. The splitting of the corresponding multiplet energy levels was visible in lower structure symmetrization.

Conclusions
By taking into consideration lattice relaxation, CDC, and CC, the multiplet energies of Mn 4+ doped K 2 XF 6 crystals were qualitatively estimated. The fluoride compounds exhibited here were suitable materials to be used as red phosphors for white LEDs since the Mn 4+ impurities in these hosts were emitted at approximately 620−630 nm (the proper spectral range to obtain a "warm" white light from the LEDs). In addition, they appeared to be thermally stable since they are known as commercial red phosphors (especially K 2 SiF 6 : Mn 4+ ). We found that the Mn-F bond length dropped, yet the U-and Y-band energies increased. By contrasting various fluoride crystal symmetry polytypes, the impact of lower symmetry was explored. The Mn-F bond length decreased, while the absorption energies of 4 A 2g → 4 T 2g ( 4 F) and 4 A 2g → 4 T 1g ( 4 F) increased, yet the 2 E g → 4 A 2g emission energy decreased. A reduced Coulomb integral result was produced when symmetry was low. It followed that a decrease in electron-electron repulsion was the cause of the declining trend in R-line energy. The CC factor c dominated the R-line energy, while the Mn-F bond length and crystal field splitting was principally responsible for the U-and Y-band energies. We discovered that low symmetry decreased the attraction between the electrons.